SMPs and Grade Level Descriptors
The Standards for Mathematical Practice (MP) are developed throughout each grade and, together with the content
standards, prescribe that students experience mathematics as a rigorous, coherent, useful and logical subject that
makes use of their ability to make sense of mathematics. The MP standards represent a picture of what it looks like for
students to understand and do mathematics in the classroom and should be integrated into every mathematics lesson
for all students.
Sixth Grade
1 Make sense of problems and persevere in solving them.
In grade 6, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world
problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for
efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to
solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”
2 Reason abstractly and quantitatively.
In grade 6, students represent a wide variety of real world contexts through the use of real numbers and variables in
mathematical expressions, equations, and inequalities. Students contextualize to understand the meaning of the number or
variable as related to the problem and decontextualize to manipulate symbolic representations by applying properties of
operations.
3 Construct viable arguments and critique the reasoning of others.
In grade 6, students construct arguments using verbal or written explanations accompanied by expressions, equations,
inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). They further refine
their mathematical communication skills through mathematical discussions in which they critically evaluate their own thinking
and the thinking of other students. They pose questions like “How did you get that?”, “Why is that true?” “Does that always
work?” They explain their thinking to others and respond to others’ thinking.
4 Model with mathematics.
In grade 6, students model problem situations symbolically, graphically, tabularly, and contextually. Students form expressions,
equations, or inequalities from real world contexts and connect symbolic and graphical representations. Students begin to
explore covariance and represent two quantities simultaneously. Students use number lines to compare numbers and represent
inequalities. They use measures of center and variability and data displays (i.e. box plots and histograms) to draw inferences
about and make comparisons between data sets. Students need many opportunities to connect and explain the connections
between the different representations. They should be able to use all of these representations as appropriate to a problem
context.
5 Use appropriate tools strategically.
Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when
certain tools might be helpful. For instance, students in grade 6 may decide to represent similar data sets using dot plots with the
same scale to visually compare the center and variability of the data. Additionally, students might use physical objects or
applets to construct nets and calculate the surface area of three-dimensional figures.
6 Attend to precision.
In grade 6, students continue to refine their mathematical communication skills by using clear and precise language in their
discussions with others and in their own reasoning. Students use appropriate terminology when referring to rates, ratios,
geometric figures, data displays, and components of expressions, equations or inequalities.
7 Look for and make use of structure.
Students routinely seek patterns or structures to model and solve problems. For instance, students recognize patterns that exist
in ratio tables recognizing both the additive and multiplicative properties. Students apply properties to generate equivalent
expressions
(i.e. 6 + 2x = 3 (2 + x) by distributive property) and solve equations (i.e. 2c + 3 = 15, 2c = 12 by subtraction property of
equality), c=6 by division property of equality). Students compose and decompose two- and three-dimensional figures to solve
real world problems involving area and volume.
8 Look for and express regularity in repeated reasoning.
In grade 6, students use repeated reasoning to understand algorithms and make generalizations about patterns. During multiple
opportunities to solve and model problems, they may notice that a/b ÷ c/d = ad/bc and construct other examples and models that
confirm their generalization. Students connect place value and their prior work with operations to understand algorithms to
fluently divide multi-digit numbers and perform all operations with multi-digit decimals. Students informally begin to make
connections between covariance, rates, and representations showing the relationships between quantities.
7th Grade
1. Make sense of problems and persevere in solving them.
In grade 7, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world
problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for
efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to
solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”
2. Reason abstractly and quantitatively. In grade 7, students represent a wide variety of real world contexts through the use of
real numbers and variables in mathematical expressions, equations, and inequalities. Students contextualize to understand the
meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic representations by
applying properties of operations.
3. Construct viable arguments and critique the reasoning of others.
In grade 7, students construct arguments using verbal or written explanations accompanied by expressions, equations,
inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). They further refine
their mathematical communication skills through mathematical discussions in which they critically evaluate their own thinking
and the thinking of other students. They pose questions like “How did you get that?”, “Why is that true?” “Does that always
work?” They explain their thinking to others and respond to others’ thinking.
4. Model with mathematics.
In grade 7, students model problem situations symbolically, graphically, tabularly, and contextually. Students form expressions,
equations, or inequalities from real world contexts and connect symbolic and graphical representations. Students explore
covariance and represent two quantities simultaneously. They use measures of center and variability and data displays (i.e. box
plots and histograms) to draw inferences, make comparisons and formulate predictions. Students use experiments or simulations
to generate data sets and create probability models. Students need many opportunities to connect and explain the connections
between the different representations. They should be able to use all of these representations as appropriate to a problem
context.
5. Use appropriate tools strategically.
Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when
certain tools might be helpful. For instance, students in grade 7 may decide to represent similar data sets using dot plots with the
same scale to visually compare the center and variability of the data. Students might use physical objects or applets to generate
probability data and use graphing calculators or spreadsheets to manage and represent data in different forms.
6. Attend to precision.
In grade 7, students continue to refine their mathematical communication skills by using clear and precise language in their
discussions with others and in their own reasoning. Students define variables, specify units of measure, and label axes
accurately. Students use appropriate terminology when referring to rates, ratios, probability models, geometric figures, data
displays, and components of expressions, equations or inequalities.
7. Look for and make use of structure.
Students routinely seek patterns or structures to model and solve problems. For instance, students recognize patterns that exist
in ratio tables making connections between the constant of proportionality in a table with the slope of a graph. Students apply
properties to generate equivalent expressions (i.e. 6 + 2x = 3 (2 + x) by distributive property) and
solve equations (i.e. 2c + 3 = 15, 2c = 12 by subtraction property of equality), c = 6 by division property of equality). Students
compose and decompose two‐ and three‐dimensional figures to solve real world problems involving scale drawings, surface
area, and volume. Students examine tree diagrams or systematic lists to determine the sample space for compound events and
verify that they have listed all possibilities.
8. Look for and express regularity in repeated reasoning.
In grade 7, students use repeated reasoning to understand algorithms and make generalizations about patterns. During multiple
opportunities to solve and model problems, they may notice that a/b ÷ c/d = ad/bc and construct other examples and models that
confirm their generalization. They extend their thinking to include complex fractions and rational numbers. Students formally
begin to make connections between covariance, rates, and representations showing the relationships between quantities. They
create, explain, evaluate, and modify probability models to describe simple and compound events.
8th Grade
1 Make sense of problems and persevere in solving them.
In grade 8, students solve real world problems through the application of algebraic and geometric concepts. Students seek
the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking
themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the
problem in a different way?”
2 Reason abstractly and quantitatively.
In grade 8, students represent a wide variety of real world contexts through the use of real numbers and variables in
mathematical expressions, equations, and inequalities. They examine patterns in data and assess the degree of linearity of
functions. Students contextualize to understand the meaning of the number or variable as related to the problem and
decontextualize to manipulate symbolic representations by applying properties of operations.
3 Construct viable arguments and critique the reasoning of others.
In grade 8, students construct arguments using verbal or written explanations accompanied by expressions, equations,
inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). They further
refine their mathematical communication skills through mathematical discussions in which they critically evaluate their
own thinking and the thinking of other students. They pose questions like “How did you get that?”, “Why is that true?”
“Does that always work?” They explain their thinking to others and respond to others’ thinking.
4 Model with mathematics.
In grade 8, students model problem situations symbolically, graphically, tabularly, and contextually. Students form
expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations.
Students solve systems of linear equations and compare properties of functions provided in different forms. Students use
scatterplots to represent data and describe associations between variables. Students need many opportunities to connect
and explain the connections between the different representations. They should be able to use all of these representations
as appropriate to a problem context.
5 Use appropriate tools strategically.
Students consider available tools (including estimation and technology) when solving a mathematical problem and decide
when certain tools might be helpful. For instance, students in grade 8 may translate a set of data given in tabular form to a
graphical representation to compare it to another data set. Students might draw pictures, use applets, or write equations to
show the relationships between the angles created by a transversal.
6 Attend to precision.
In grade 8, students continue to refine their mathematical communication skills by using clear and precise language in
their discussions with others and in their own reasoning. Students use appropriate terminology when referring to the
number system, functions, geometric figures, and data displays.
7 Look for and make use of structure.
Students routinely seek patterns or structures to model and solve problems. In grade 8, students apply properties to
generate equivalent expressions and solve equations. Students examine patterns in tables and graphs to generate equations
and describe relationships. Additionally, students experimentally verify the effects of transformations and describe them
in terms of congruence and similarity.
8 Look for and express regularity in repeated reasoning.
In grade 8, students use repeated reasoning to understand algorithms and make generalizations about patterns. Students
use iterative processes to determine more precise rational approximations for irrational numbers. During multiple
opportunities to solve and model problems, they notice that the slope of a line and rate of change are the same value.
Students flexibly make connections between covariance, rates, and representations showing the relationships between
quantities.